With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations. As shown in [Gri13] PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point. The resulting PL functions are obtained quite naturally in what we call the abs-normal form, a variant of the state representation proposed by Bokhoven in his dissertation [vB81]. Apart from the tradition of PL modelling by electrical engineers, which dates back to the Master thesis of Thomas Stern [Ste56] in 1956, we take into account more recent results on linear complementarity problems and semi-smooth equations originating in the optimization community [CPS92, Sch12, FP03]. We analyze simultaneously the original PL problem (OPL) in abs-normal form and a corresponding complementary system (CPL), which is closely related to the absolute value equation (AVE) studied by Mangasarian et al [MM06] and a corresponding linear complementarity problem (LCP). We show that the CPL, like KKT conditions and other simply switched systems, cannot be open without being injective. Hence some of the intriguing PL structure described by Scholtes in [Sch12] is lost in the transformation from OPL to CPL. To both problems one may apply Newton variants with appropriate generalized Jacobians directly computable from the abs-normal representation. Alternatively, the CPL can be solved by Bokhoven's modulus method and related fixed point iterations. We compile the properties of the various schemes and highlight the connection to the properties of the Schur complement matrix, in particular its signed real spectral radius as analyzed by Rump in [Rum97]. Numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized Newton variants remain to be realized.

Let S be a real n × n matrix, z,ĉ ∈ R n , and |z| the componentwise modulus of z. Then the piecewise linear equation system z − S|z| =ĉ is called an absolute value equation (AVE). It has been proven to be equivalent to the general linear complementarity problem, which means that it is NP hard in general.We will show that for several system classes the AVE essentially retains the good natured solvability properties of regular linear systems. I.e., it can be solved directly by a slightly modified Gaussian elimination that we call the signed Gaussian elimination. For dense matrices S this algorithm has the same operations count as the classical Gaussian elimination with symmetric pivoting. For tridiagonal systems in n variables its computational cost is roughly that of sorting n floating point numbers. The sharpness of the proposed restrictions on S will be established.

International audienceIt follows from the well known min/max representation given by Scholtes in his recent Springer book, that all piecewise linear continuous functions $$y = F(x) : \mathbb {R}^n \rightarrow \mathbb {R}^m$$ can be written in a so-called abs-normal form. This means in particular, that all nonsmoothness is encapsulated in $$s$$ absolute value functions that are applied to intermediate switching variables $$z_i$$ for $$i=1, \ldots ,s$$. The relation between the vectors $$x, z$$, and $$y$$ is described by four matrices $$Y, L, J$$, and $$Z$$, such that $$ \left[ \begin{array}{c} z \\ y \end{array}\right] = \left[ \begin{array}{c} c \\ b \end{array}\right] + \left[ \begin{array}{cc} Z &{} L \\ J &{} Y \end{array}\right] \left[ \begin{array}{c} x \\ |z |\end{array}\right] $$This form can be generated by ADOL-C or other automatic differentation tools. Here $$L$$ is a strictly lower triangular matrix, and therefore $$ z_i$$ can be computed successively from previous results. We show that in the square case $$n=m$$ the system of equations $$F(x) = 0$$ can be rewritten in terms of the variable vector $$z$$ as a linear complementarity problem (LCP). The transformation itself and the properties of the LCP depend on the Schur complement $$S = L - Z J^{-1} Y$$

It is shown how piecewise differentiable functions F : R n → R m that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample pointsx andx. We show that the discrepancy between function and model at any point x is of the bilinear order O( x −x x −x ). As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

A standard theorem in nonsmooth analysis states that a piecewise affine function F : R n → R n is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero determinant sign. In this note we prove that surjectivity already follows from coherent orientation of the selection functions which are active on the unbounded sets of a polyhedral subdivision of the domain corresponding to F . A side bonus of the argumentation is a short proof of the classical statement that an injective piecewise affine function is coherently oriented.